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Theorem funssxp 5482
Description: Two ways of specifying a partial function from  A to  B. (Contributed by NM, 13-Nov-2007.)
Assertion
Ref Expression
funssxp  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)

Proof of Theorem funssxp
StepHypRef Expression
1 funfn 5362 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
21biimpi 186 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
3 rnss 4986 . . . . . 6  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
4 rnxpss 5187 . . . . . 6  |-  ran  ( A  X.  B )  C_  B
53, 4syl6ss 3267 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
62, 5anim12i 549 . . . 4  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F  Fn  dom  F  /\  ran  F  C_  B )
)
7 df-f 5338 . . . 4  |-  ( F : dom  F --> B  <->  ( F  Fn  dom  F  /\  ran  F 
C_  B ) )
86, 7sylibr 203 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  F : dom  F --> B )
9 dmss 4957 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  dom  ( A  X.  B ) )
10 dmxpss 5186 . . . . 5  |-  dom  ( A  X.  B )  C_  A
119, 10syl6ss 3267 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  A )
1211adantl 452 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  dom  F 
C_  A )
138, 12jca 518 . 2  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
14 ffun 5471 . . . 4  |-  ( F : dom  F --> B  ->  Fun  F )
1514adantr 451 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  Fun  F )
16 fssxp 5480 . . . 4  |-  ( F : dom  F --> B  ->  F  C_  ( dom  F  X.  B ) )
17 xpss1 4874 . . . 4  |-  ( dom 
F  C_  A  ->  ( dom  F  X.  B
)  C_  ( A  X.  B ) )
1816, 17sylan9ss 3268 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  F  C_  ( A  X.  B
) )
1915, 18jca 518 . 2  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  ( Fun  F  /\  F  C_  ( A  X.  B
) ) )
2013, 19impbii 180 1  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    C_ wss 3228    X. cxp 4766   dom cdm 4768   ran crn 4769   Fun wfun 5328    Fn wfn 5329   -->wf 5330
This theorem is referenced by:  elpm2g  6872  volf  18986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-dm 4778  df-rn 4779  df-fun 5336  df-fn 5337  df-f 5338
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